Determining Differences Between Groups

1. Determining Differences Between Groups T-tests ANOVA

2. Some background…... • Researchers want to know the characteristics of a large group (i.e. all women). IMPOSSIBLE!! • However a representative “sample” can be selected • Results are then inferred to apply to the population.

3. Background • This is called inferential statistics.

4. Comparing Means • Standard Error of the Mean= SEM • The standard deviation of a sampling distribution of means • SD divided by the square root of “n-1”

5. Normal Curves • Just like with scores and SD, 68% of sample means from a population fall within +1SEM of the population mean

6. % of Sample Means in 1 SEM 1 SEM = 68% of sample means

7. 2 SEM? 2 SEM = 95% of sample means

8. Example • Mean of a population is 50 and the SEM is 5 • Very likely to select a sample with a mean of 45 but very unlikely to select a sample with a mean that is 35.

9. Comparing Means • Most of the time researchers want to compare 2 or more samples • IE, difference in reading ability between 3rd grade boys and girls.

10. More background • Initial stats approach is to assume that groups are NOT different, then test statistically. • This is the NULL HYPOTHESIS • “ Boys and Girls reading abilities do not differ” • “Male heights are equal to female heights”

11. Comparing Means • Obviously two means from a sample will never be exactly alike. • We want to know if they are different due to chance or is there a real reason.

12. Comparing means • We can obtain the probability that two means will differ simply due to chance. • If the probability is small we reject the null and accept the research hypothesis: “there is a difference between male and female height”

13. In other words………... • If the probability is less than .05, groups are considered different • This is known as the Alpha level.

14. Statistical Tests • To compare two means we use a t-test • We can use the normal curve to determine the a difference between two means is likely.

15. Types of t-tests • Independent (unpaired): compare two different groups • Dependent (paired): same subjects in both groups. Some treatment has been applied.

16. Grab a representative sample Men Population Women

17. Compare Height Means and SD W M Unpaired T-Test

18. The t-test determines the probability of these groups being the same

19. Results • “The probability that the mean difference we observe between the male and female heights is due to chance is .034, thus we reject the null and accept the research hypothesis”

20. Other ways of saying it…. • The probability of obtaining such an outcome is only 3.4 in 100 • Our results are likely to occur by chance less than 3.4 percent of the time.

21. P-Value • P less than .05 is significant (reject null, accept research) • P greater than .05 is not significant (accept null, reject research) • The p value is .05 a lot of times but you may see .01, .10, or an adjusted p value sometimes as you read.

22. By the way……….. • The significance of our example isn’t very meaningful since we probably already knew that males on average are significantly taller than females and there are reasons for our that.

23. Directional Hypotheses • Maybe our research hypothesis should have been: “Males are taller than females” • Before: “Male and female heights are different” (two-tailed)

24. Tails • One-tailed T-test • We decide ahead of time that we will subtract the mean of the females from the males (thus expect a positive difference) • If we get a negative difference it will not support our research hypothesis and we would reject the null

25. Quick Assignment • Using the class data spreadsheet, perform a paired (dependent) t-test for thigh vs. triceps

26. Assignment • Use the t-test excel spreadsheet on my web page t-test

27. Assignment1 • ACT1=no test prep • ACT2=test preparation • These are two separate sets of subject. • Perform the appropriate t-test

28. Assignment2 • Height1 = height in third grade • Height2 = height in sixth grade • Performed on same set of subjects

29. Assignment3 • Attitude 1=attitude toward school at beginning of the year • Attitude 2=attitude toward school at end of the year • Performed on same group of subjects

30. ANOVA • Again, two types • Repeated Measures • Factorial • More than two means to compare

31. Repeated Measures (ANOVA) • Used when performing repeated measures on the same group of subjects (analogous to a dependent t-test) • I.e. changes in strength over 12 weeks of training…………..

32. Subjects • 20 subjects have bench press 1RM assessed at 1, 4, 8 and 12 weeks • One way ANOVA used to determine changes across time • If significant, then post-hoc tests (Tukey, Scheffe)

33. 1 4 8 12 1RM changes p = .02

34. 1 4 8 12 “post hoc” t=tests Wk 4 > 1 p= .03 Wk 8 > 1 p = .03 Wk 12 > 1 p = .01 Wk 12 > 4 p = .03 Wk 8 > 4 p = .02 Wk 12 > 8 p = .15 Conclude??

35. Factorial ANOVA • Used when have more than two groups of subjects (analogous to independent t-test) • Comparing three different types of teaching styles on 3 different groups of 3rd graders

36. Higher level ANOVAS • Previous examples were one-way ANOVAs (only one independent variable) • Can have studies with multiple independent variables

37. Variables • Dependent and Independent Variables • Dependent – what you are measuring or interested in. • Independent – usually categorical and what remains constant.

38. Actually……… • The training study should be a mixed factorial repeated measures ANOVA since you probably have a Control group.

39. Training Control Results Time (wks)

40. Example • Two groups (control and training) • Measured strength, EMG, MMG every 4 weeks for 12 weeks • Measured strength, EMG, MMG of both legs • Measured strength, EMG, MMG at 7 different velocities

41. Variables? • Independent? • Dependent? • A 2 x 2 x 4 x 7 mixed factorial repeated measures ANOVA for each dependent variable (otherwise would have to do a MANOVA – two or more dependent variables)

42. Other Higher Order Examples • You want to know how three separate third grade classes respond to 3 different teaching styles and you measured test scores at 3 different times during the year.

43. Answer • 3 (treatment) by 3 (week of year) factorial ANOVA (3 separate groups of subjects).

44. Example • You are following a group of 1st graders throughout their elementary years and measuring the height of both the boys and girls.

45. Answer • 6 (years of elementary) x 2 (gender) repeated measures (same group of subjects) mixed factorial ANOVA

46. Example • You are following a group of 1st graders throughout their elementary years and measuring the height (twice a year) of both the boys and girls.

47. Answer • 6 (years of elementary) x 2 (gender) x 2 (times measured) repeated measures mixed factorial ANOVA

48. Columns of Data • Year1-1, Year1-2, Year 2-1, Year2-2, Year3-1…………… • Would also have a gender column denoting which subjects are girls and which are boys (boys = 1, girls = 2)

49. Assumptions • 1. Data are normally distributed • 2. Random sampling • 3. Groups are independent of each other (random assignment into treatment groups) • 4. In studies using multiple samples, the populations represented are assumed to be equally variable

50. Non-parametric Tests • Mann-Whitney • Kruskal-Wallis • Sign Test • Friedman • Chi-Square